3.2755 \(\int (c x)^m (a+b x^n)^2 \, dx\)

Optimal. Leaf size=64 \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b x^{n+1} (c x)^m}{m+n+1}+\frac{b^2 x^{2 n+1} (c x)^m}{m+2 n+1} \]

[Out]

(2*a*b*x^(1 + n)*(c*x)^m)/(1 + m + n) + (b^2*x^(1 + 2*n)*(c*x)^m)/(1 + m + 2*n) + (a^2*(c*x)^(1 + m))/(c*(1 +
m))

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Rubi [A]  time = 0.0334044, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {270, 20, 30} \[ \frac{a^2 (c x)^{m+1}}{c (m+1)}+\frac{2 a b x^{n+1} (c x)^m}{m+n+1}+\frac{b^2 x^{2 n+1} (c x)^m}{m+2 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x^n)^2,x]

[Out]

(2*a*b*x^(1 + n)*(c*x)^m)/(1 + m + n) + (b^2*x^(1 + 2*n)*(c*x)^m)/(1 + m + 2*n) + (a^2*(c*x)^(1 + m))/(c*(1 +
m))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (c x)^m \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 (c x)^m+2 a b x^n (c x)^m+b^2 x^{2 n} (c x)^m\right ) \, dx\\ &=\frac{a^2 (c x)^{1+m}}{c (1+m)}+(2 a b) \int x^n (c x)^m \, dx+b^2 \int x^{2 n} (c x)^m \, dx\\ &=\frac{a^2 (c x)^{1+m}}{c (1+m)}+\left (2 a b x^{-m} (c x)^m\right ) \int x^{m+n} \, dx+\left (b^2 x^{-m} (c x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac{2 a b x^{1+n} (c x)^m}{1+m+n}+\frac{b^2 x^{1+2 n} (c x)^m}{1+m+2 n}+\frac{a^2 (c x)^{1+m}}{c (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0366014, size = 47, normalized size = 0.73 \[ x (c x)^m \left (\frac{a^2}{m+1}+\frac{2 a b x^n}{m+n+1}+\frac{b^2 x^{2 n}}{m+2 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(a + b*x^n)^2,x]

[Out]

x*(c*x)^m*(a^2/(1 + m) + (2*a*b*x^n)/(1 + m + n) + (b^2*x^(2*n))/(1 + m + 2*n))

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Maple [C]  time = 0.047, size = 234, normalized size = 3.7 \begin{align*}{\frac{ \left ({b}^{2}{m}^{2} \left ({x}^{n} \right ) ^{2}+{b}^{2}mn \left ({x}^{n} \right ) ^{2}+2\,ab{m}^{2}{x}^{n}+4\,abmn{x}^{n}+2\,m{b}^{2} \left ({x}^{n} \right ) ^{2}+{b}^{2}n \left ({x}^{n} \right ) ^{2}+{a}^{2}{m}^{2}+3\,{a}^{2}mn+2\,{a}^{2}{n}^{2}+4\,mab{x}^{n}+4\,abn{x}^{n}+{b}^{2} \left ({x}^{n} \right ) ^{2}+2\,m{a}^{2}+3\,{a}^{2}n+2\,a{x}^{n}b+{a}^{2} \right ) x}{ \left ( 1+m \right ) \left ( m+n+1 \right ) \left ( 1+m+2\,n \right ) }{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( icx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( icx \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( c \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(a+b*x^n)^2,x)

[Out]

x*(b^2*m^2*(x^n)^2+b^2*m*n*(x^n)^2+2*a*b*m^2*x^n+4*a*b*m*n*x^n+2*m*b^2*(x^n)^2+b^2*n*(x^n)^2+a^2*m^2+3*a^2*m*n
+2*a^2*n^2+4*m*a*b*x^n+4*a*b*n*x^n+b^2*(x^n)^2+2*m*a^2+3*a^2*n+2*a*x^n*b+a^2)/(1+m)/(m+n+1)/(1+m+2*n)*exp(1/2*
m*(-I*csgn(I*c*x)^3*Pi+I*csgn(I*c*x)^2*csgn(I*c)*Pi+I*csgn(I*c*x)^2*csgn(I*x)*Pi-I*csgn(I*c*x)*csgn(I*c)*csgn(
I*x)*Pi+2*ln(x)+2*ln(c)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(a+b*x^n)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.3744, size = 417, normalized size = 6.52 \begin{align*} \frac{{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2} +{\left (b^{2} m + b^{2}\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + 2 \,{\left (a b m^{2} + 2 \, a b m + a b + 2 \,{\left (a b m + a b\right )} n\right )} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} +{\left (a^{2} m^{2} + 2 \, a^{2} n^{2} + 2 \, a^{2} m + a^{2} + 3 \,{\left (a^{2} m + a^{2}\right )} n\right )} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(a+b*x^n)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 2*b^2*m + b^2 + (b^2*m + b^2)*n)*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 2*(a*b*m^2 + 2*a*b*m + a*b +
2*(a*b*m + a*b)*n)*x*x^n*e^(m*log(c) + m*log(x)) + (a^2*m^2 + 2*a^2*n^2 + 2*a^2*m + a^2 + 3*(a^2*m + a^2)*n)*x
*e^(m*log(c) + m*log(x)))/(m^3 + 2*(m + 1)*n^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*n + 3*m + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(a+b*x**n)**2,x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.18254, size = 828, normalized size = 12.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(a+b*x^n)^2,x, algorithm="giac")

[Out]

(b^2*m^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + b^2*m*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 2*a*b*m^2*x*x^n*e^(m*
log(c) + m*log(x)) + b^2*m^2*x*x^n*e^(m*log(c) + m*log(x)) + 4*a*b*m*n*x*x^n*e^(m*log(c) + m*log(x)) + b^2*m*n
*x*x^n*e^(m*log(c) + m*log(x)) + a^2*m^2*x*e^(m*log(c) + m*log(x)) + 2*a*b*m^2*x*e^(m*log(c) + m*log(x)) + b^2
*m^2*x*e^(m*log(c) + m*log(x)) + 3*a^2*m*n*x*e^(m*log(c) + m*log(x)) + 4*a*b*m*n*x*e^(m*log(c) + m*log(x)) + b
^2*m*n*x*e^(m*log(c) + m*log(x)) + 2*a^2*n^2*x*e^(m*log(c) + m*log(x)) + 2*b^2*m*x*x^(2*n)*e^(m*log(c) + m*log
(x)) + b^2*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 4*a*b*m*x*x^n*e^(m*log(c) + m*log(x)) + 2*b^2*m*x*x^n*e^(m*lo
g(c) + m*log(x)) + 4*a*b*n*x*x^n*e^(m*log(c) + m*log(x)) + b^2*n*x*x^n*e^(m*log(c) + m*log(x)) + 2*a^2*m*x*e^(
m*log(c) + m*log(x)) + 4*a*b*m*x*e^(m*log(c) + m*log(x)) + 2*b^2*m*x*e^(m*log(c) + m*log(x)) + 3*a^2*n*x*e^(m*
log(c) + m*log(x)) + 4*a*b*n*x*e^(m*log(c) + m*log(x)) + b^2*n*x*e^(m*log(c) + m*log(x)) + b^2*x*x^(2*n)*e^(m*
log(c) + m*log(x)) + 2*a*b*x*x^n*e^(m*log(c) + m*log(x)) + b^2*x*x^n*e^(m*log(c) + m*log(x)) + a^2*x*e^(m*log(
c) + m*log(x)) + 2*a*b*x*e^(m*log(c) + m*log(x)) + b^2*x*e^(m*log(c) + m*log(x)))/(m^3 + 3*m^2*n + 2*m*n^2 + 3
*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)